# If we find a Taylor Series for a function and then evaluate were it converges must it represent a power series?

I am looking to clarify the relationship between a power series and a Taylor series. I realize that if I discover that a function can be represented by a power series then it seems straight forward I can work out the Taylor series.

First question is: Since I know the radius of convergence of the power series I assume I can apply the same radius to the Taylor Series where the center point would be located in the radius?

Second. If I generate a Taylor series from a function I know that it may not necessarily mean a power series exists for that function but if I find the radius of convergence of the Taylor series does that mean then that a power series does exist for that radius? I realize at least one point of convergence must exist for any power series but is there some other factor that determines if my Taylor series will have a power series associated with it?

Although the main question is question two which is embedded in the title of my post but I feel the first question needs a little clarification for me as well ...Thank you

• Your question does need some clarification. Why do you distinguish power series and Taylor series? – i707107 Feb 25 '18 at 17:36
• yes, you have a good point. Perhaps you are saying a Taylor series is just a type of power series? It would be nice if I could find a simple differences between power series and what seems to be special fabrications of power series such as Taylor series, geometric series and now I am working on binomial series and that seems to me to be a special case of a power series also.....the confusion would be cleared if the book would present them in one page and highlight the difference instead of me trying to unify the concepts ....my memory is so short I forget the original object to unify ;-) – Sedumjoy Feb 26 '18 at 16:25